Optimal. Leaf size=38 \[ \frac{\left (a+b x^4\right )^{7/4}}{7 b^2}-\frac{a \left (a+b x^4\right )^{3/4}}{3 b^2} \]
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Rubi [A] time = 0.0211263, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{\left (a+b x^4\right )^{7/4}}{7 b^2}-\frac{a \left (a+b x^4\right )^{3/4}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^7}{\sqrt [4]{a+b x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{\sqrt [4]{a+b x}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{a}{b \sqrt [4]{a+b x}}+\frac{(a+b x)^{3/4}}{b}\right ) \, dx,x,x^4\right )\\ &=-\frac{a \left (a+b x^4\right )^{3/4}}{3 b^2}+\frac{\left (a+b x^4\right )^{7/4}}{7 b^2}\\ \end{align*}
Mathematica [A] time = 0.0116985, size = 28, normalized size = 0.74 \[ \frac{\left (a+b x^4\right )^{3/4} \left (3 b x^4-4 a\right )}{21 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-3\,b{x}^{4}+4\,a}{21\,{b}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.999151, size = 41, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{4} + a\right )}^{\frac{7}{4}}}{7 \, b^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}} a}{3 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67067, size = 59, normalized size = 1.55 \begin{align*} \frac{{\left (3 \, b x^{4} - 4 \, a\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{21 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.73808, size = 44, normalized size = 1.16 \begin{align*} \begin{cases} - \frac{4 a \left (a + b x^{4}\right )^{\frac{3}{4}}}{21 b^{2}} + \frac{x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{7 b} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 \sqrt [4]{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09608, size = 39, normalized size = 1.03 \begin{align*} \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} - 7 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a}{21 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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